Function 3 roots of x. Power function and roots - definition, properties and formulas

Instead of introducing

The use of modern technologies (CTE) and teaching aids (multimedia board) in lessons helps the teacher plan and conduct effective lessons, create conditions for students to consciously understand, memorize and practice skills.

The lesson turns out to be dynamic and interesting if during training session combine different forms of training.

In modern didactics, there are four general organizational forms of training:

• individually mediated;
• steam room;
• group;

collective (in shift pairs). (Dyachenko V.K. Modern didactics. - M.: Public education, 2005).

In a traditional lesson, as a rule, only the first three organizational forms of teaching listed above are used. The collective form of teaching (work in pairs in shifts) is practically not used by the teacher. However, this organizational form of training makes it possible for the team to train everyone and everyone to actively participate in the training of others. The collective form of training is leading in CSR technology.

One of the most common methods of collective learning technology is the “Mutual Training” technique.

This “magic” technique is good in any subject and in any lesson. The purpose is training.

Training is the successor of self-control; it helps the student to establish contact with the subject of study, making it easier to find the right steps and actions. Through training in the acquisition, consolidation, regrouping, revision, and application of knowledge, a person’s cognitive abilities develop. (Yanovitskaya E.V. How to teach and learn in lesson like this to want to learn. Album-reference book. – St. Petersburg: Educational projects, M.: Publisher A.M. Kushnir, 2009.-P.14;131)

It will help you quickly repeat a rule, remember the answers to the questions you have studied, and consolidate the necessary skill. The optimal time to work using the method is 5-10 minutes. As a rule, work on training cards is carried out during oral calculation, that is, at the beginning of the lesson, but at the discretion of the teacher it can be carried out at any stage of the lesson, depending on its goals and structure. A training card can contain from 5 to 10 simple examples (questions, tasks). Each student in the class receives a card. The cards are different for everyone or different for everyone in the “combined squad” (children sitting on the same row). A combined detachment (group) is a temporary cooperation of students formed to perform a specific educational task. (Yalovets T.V. Technology of a collective method of teaching in teacher training: Educational and methodological manual. - Novokuznetsk: IPK Publishing House, 2005. - P. 122)

Lesson project on the topic “Function y=, its properties and graph”

In the lesson project, the topic of which is: “ Function y=, its properties and graph” The use of mutual training techniques in combination with the use of traditional and multimedia teaching tools is presented.

Lesson topic: “ Function y=, its properties and graph”

Goals:

• preparation for the test;
• testing knowledge of all properties of a function and the ability to build graphs of functions and read their properties.

Tasks: subject level:

supra-subject level:

• learn to analyze graphic information;
• practice the ability to conduct dialogue;
• develop the ability to work with an interactive whiteboard using the example of working with graphs.

Let us reveal in more detail the content of each stage.

1. Teacher Information Input (TII) includes Organizing time; articulating the topic, purpose and lesson plan; showing a sample of pair work using the mutual training method.

Demonstration of a sample of work in pairs by students at this stage of the lesson is advisable for repeating the algorithm of work of the methodology we need, because at the next stage of the lesson, all work is planned on it cool team. At the same time, you can name the errors in working with the algorithm (if there were any), as well as evaluate the work of these students.

2. Updating of basic knowledge is carried out in shift pairs using the method of mutual training.

The methodology algorithm includes individual, pair (static pairs) and collective (shift pairs) organizational forms of training.

Individual: everyone who receives the card gets acquainted with its contents (reads the questions and answers on the back of the card).

• first(in the role of the “trainee”) reads the task and answers the questions on the partner’s card;
• second(in the role of “coach”) – checks the correctness of the answers on the back of the card;
• work similarly on another card, changing roles;
• make a mark on an individual sheet and exchange cards;
• move to a new couple.

Collective:

• in the new pair they work like in the first; transition to a new pair, etc.

The number of transitions depends on the time allocated by the teacher for this stage of the lesson, on the diligence and speed of comprehension of each student and on the partners in joint work.

After working in pairs, students make marks on their record sheets, and the teacher conducts a quantitative and qualitative analysis of the work.

The accounting sheet may look like this:

Ivanov Petya 7 “b” grade

3. Introduction to the topic “Function y=, its properties and graph” is carried out by the teacher in the form of a presentation using multimedia learning tools (Appendix 4). On the one hand, this is a version of clarity that is understandable to modern students, on the other hand, it saves time on explaining new material.

4. Consolidation of newly learned and already covered material on the topic “Function ” organized in two versions, using traditional teaching tools (blackboard, textbook) and innovative ones (interactive whiteboard).

First, several tasks from the textbook are offered to consolidate the newly learned material. The textbook used for teaching is used. Work is carried out simultaneously with the whole class. In this case, one student completes task “a” - on a traditional board; the other is task “b” on interactive whiteboard, the rest of the students write down the solutions to the same tasks in a notebook and compare their solution with the solution presented on the boards. Next, the teacher evaluates the students’ work at the board.

Then, to more quickly consolidate the studied material on the topic “Function”, frontal work with an interactive whiteboard is proposed, which can be organized as follows:

• the task and schedule appear on the interactive board;
• a student who wants to answer goes to the board, performs the necessary constructions and voices the answer;
• a new task and a new schedule appear on the board;
• Another student comes out to answer.

Thus, in a short period of time, it is possible to solve quite a lot of tasks and evaluate student answers. Some tasks of interest (similar to tasks from the upcoming test work), can be recorded in a notebook.

5. At the self-control stage, students are offered a test followed by self-test (Appendix 3).

Literature

1. Dyachenko, V.K. Modern didactics [Text] / V.K. Dyachenko - M.: Public education, 2005.
2. Yalovets, T.V. Technology of a collective method of teaching in teacher training: Educational and methodological manual[Text] / T.V. Yalovets. – Novokuznetsk: IPK Publishing House, 2005.
3. Yanovitskaya, E.V. How to teach and learn in a lesson so that you want to learn. Reference album [Text] / E.V. Yanovitskaya. – St. Petersburg: Educational projects, M.: Publisher A.M. Kushnir, 2009.

Which is equal to $a.$ In other words, this is the solution to the equation $x^3\; =\; a$(usually real solutions are meant).

Real root

Demonstrative form

The root of complex numbers can be defined as follows:

$x^(1/3)\; =\; \backslash exp\; (\backslash tfrac13\; \backslash ln(x))$

If you imagine $x$ How

$x\; =\; r\backslash exp(i\backslash theta)$

then the formula for a cubic number is:

$\backslash sqrt(x)\; =\; \backslash sqrt(r)\backslash exp\; (\backslash tfrac13\; i\backslash theta).$

This geometrically means that in polar coordinates we take the cube root of the radius and divide the polar angle by three to determine the cube root. So if $x$ complex, then $\backslash sqrt(-8)$ will mean not $-2$, Will be $1\; +\; i\backslash sqrt(3).$

At a constant density of matter, the dimensions of two similar bodies are related to each other as the cube roots of their masses. So, if one watermelon weighs twice as much as another, then its diameter (as well as its circumference) will be only a little more than a quarter (26%) larger than the first; and to the eye it will seem that the difference in weight is not so significant. Therefore, in the absence of scales (sale by eye), it is usually more profitable to buy a larger fruit.

Calculation methods

Column

Before starting, you need to divide the number into triplets (the integer part - from right to left, the fractional part - from left to right). When you reach the decimal point, you must add a decimal point at the end of the result.

The algorithm is as follows:

1. Find a number whose cube is smaller than the first group of digits, but when it increases by 1 it becomes larger. Write down the number you find to the right of given number. Write the number 3 below it.
2. Write the cube of the number found under the first group of numbers and subtract. Write the result after subtraction under the subtrahend. Next, take down the next group of numbers.
3. Next, we replace the found intermediate answer with the letter $a$. Calculate using the formula $$ such a number $x$ that its result is less than the lower number, but when increased by 1 it becomes larger. Write down what you find $x$ to the right of the answer. If the required accuracy is achieved, stop calculations.
4. Write down the result of the calculation under the bottom number using the formula $300\backslash times\; a^2\backslash times\; x+30\backslash times\; a\backslash times\; x^2+x^3$ and do the subtraction. Go to step 3.

see also

Write a review about the article "Cubic root"

Literature

• Korn G., Korn T. 1.3-3. Representation of sum, product and quotient. Powers and roots // Handbook of mathematics. - 4th edition. - M.: Nauka, 1978. - P. 32-33.

An excerpt characterizing the cube root

By nine o'clock in the morning, when the troops had already moved through Moscow, no one else came to ask the count's orders. Everyone who could go did so of their own accord; those who remained decided with themselves what they had to do.
The count ordered the horses to be brought in to go to Sokolniki, and, frowning, yellow and silent, with folded hands, he sat in his office.
In calm, not stormy times, it seems to every administrator that it is only through his efforts that the entire population under his control moves, and in this consciousness of his necessity, every administrator feels the main reward for his labors and efforts. It is clear that as long as the historical sea is calm, the ruler-administrator, with his fragile boat resting his pole against the ship of the people and himself moving, must seem to him that through his efforts the ship he is resting against is moving. But as soon as a storm arises, the sea becomes agitated and the ship itself moves, then delusion is impossible. The ship moves with its enormous, independent speed, the pole does not reach the moving ship, and the ruler suddenly goes from the position of a ruler, a source of strength, into an insignificant, useless and weak person.
Rastopchin felt this, and it irritated him. The police chief, who was stopped by the crowd, together with the adjutant, who came to report that the horses were ready, entered the count. Both were pale, and the police chief, reporting the execution of his assignment, said that in the count’s courtyard there was a huge crowd of people who wanted to see him.
Rastopchin, without answering a word, stood up and quickly walked into his luxurious, bright living room, walked up to the balcony door, grabbed the handle, left it and moved to the window, from which the whole crowd could be seen more clearly. A tall fellow stood in the front rows and with a stern face, waving his hand, said something. The bloody blacksmith stood next to him with a gloomy look. The hum of voices could be heard through the closed windows.
- Is the crew ready? - said Rastopchin, moving away from the window.
“Ready, your Excellency,” said the adjutant.
Rastopchin again approached the balcony door.
- What do they want? – he asked the police chief.
- Your Excellency, they say that they were going to go against the French on your orders, they shouted something about treason. But a violent crowd, your Excellency. I left by force. Your Excellency, I dare to suggest...
“If you please, go, I know what to do without you,” Rostopchin shouted angrily. He stood at the balcony door, looking out at the crowd. “This is what they did to Russia! This is what they did to me!” - thought Rostopchin, feeling an uncontrollable anger rising in his soul against someone who could be attributed to the cause of everything that happened. As often happens with hot-tempered people, anger was already possessing him, but he was looking for another subject for it. “La voila la populace, la lie du peuple,” he thought, looking at the crowd, “la plebe qu"ils ont soulevee par leur sottise. Il leur faut une victime, [“Here it is, people, these scum of the population, the plebeians, whom they raised with their stupidity! They need a victim."] - it occurred to him, looking at the tall fellow waving his hand. And for the same reason it came to his mind that he himself needed this victim, this object for his anger.
- Is the crew ready? – he asked another time.
- Ready, Your Excellency. What do you order about Vereshchagin? “He’s waiting at the porch,” answered the adjutant.
- A! - Rostopchin cried out, as if struck by some unexpected memory.
And, quickly opening the door, he stepped out onto the balcony with decisive steps. The conversation suddenly stopped, hats and caps were taken off, and all eyes rose to the count who had come out.
- Hello guys! - the count said quickly and loudly. - Thank you for coming. I’ll come out to you now, but first of all we need to deal with the villain. We need to punish the villain who killed Moscow. Wait for me! “And the count just as quickly returned to his chambers, slamming the door firmly.
A murmur of pleasure ran through the crowd. “That means he will control all the villains! And you say French... he’ll give you the whole distance!” - people said, as if reproaching each other for their lack of faith.

The basic properties of the power function are given, including formulas and properties of the roots. Derivative, integral, expansion in power series and representation through complex numbers of a power function.

Definition

Definition
Power function with exponent p is the function f (x) = x p, the value of which at point x is equal to the value of the exponential function with base x at point p.
In addition, f (0) = 0 p = 0 for p > 0 .

For natural values ​​of the exponent, the power function is the product of n numbers equal to x:
.
It is defined for all valid .

For positive rational values ​​of the exponent, the power function is the product of n roots of degree m of the number x:
.
For odd m, it is defined for all real x. For even m, the power function is defined for non-negative ones.

For negative , the power function is determined by the formula:
.
Therefore, it is not defined at the point.

For irrational values ​​of the exponent p, the power function is determined by the formula:
,
where a is an arbitrary positive number not equal to one: .
When , it is defined for .
When , the power function is defined for .

Continuity. A power function is continuous in its domain of definition.

Properties and formulas of power functions for x ≥ 0

Here we will consider the properties of the power function for not negative values argument x. As stated above, for certain values ​​of the exponent p, the power function is also defined for negative values ​​of x. In this case, its properties can be obtained from the properties of , using even or odd. These cases are discussed and illustrated in detail on the page "".

A power function, y = x p, with exponent p has the following properties:
(1.1) defined and continuous on the set
at ,
at ;
(1.2) has many meanings
at ,
at ;
(1.3) strictly increases with ,
strictly decreases as ;
(1.4) at ;
at ;
(1.5) ;
(1.5*) ;
(1.6) ;
(1.7) ;
(1.7*) ;
(1.8) ;
(1.9) .

Proof of properties is given on the page “Power function (proof of continuity and properties)”

Roots - definition, formulas, properties

Definition
Root of a number x of degree n is the number that when raised to the power n gives x:
.
Here n = 2, 3, 4, ... - natural number, greater than one.

You can also say that the root of a number x of degree n is the root (i.e. solution) of the equation
.
Note that the function is the inverse of the function.

Square root of x is a root of degree 2: .

Cube root from number x is a root of degree 3: .

Even degree

For even powers n = 2 m, the root is defined for x ≥ 0 . A formula that is often used is valid for both positive and negative x:
.
For square root:
.

The order in which the operations are performed is important here - that is, first squaring is performed, resulting in a non-negative number, and then the root is extracted from it (you can extract from a non-negative number Square root). If we changed the order: , then for negative x the root would be undefined, and with it the entire expression would be undefined.

Odd degree

For odd powers, the root is defined for all x:
;
.

Properties and formulas of roots

The root of x is a power function:
.
When x ≥ 0 the following formulas apply:
;
;
, ;
.

These formulas can also be applied for negative values ​​of variables. You just need to make sure that the radical expression of even powers is not negative.

Private values

The root of 0 is 0: .
Root 1 is equal to 1: .
The square root of 0 is 0: .
The square root of 1 is 1: .

Example. Root of roots

Let's look at an example of a square root of roots:
.
Let's transform the inner square root using the formulas above:
.
Now let's transform the original root:
.
So,
.

y = x p for different values ​​of the exponent p.

Here are graphs of the function for non-negative values ​​of the argument x. Graphs of a power function defined for negative values ​​of x are given on the page “Power function, its properties and graphs"

Inverse function

The inverse of a power function with exponent p is a power function with exponent 1/p.

If, then.

Derivative of a power function

Derivative of nth order:
;

Deriving formulas > > >

Integral of a power function

P ≠ - 1 ;
.

Power series expansion

At - 1 < x < 1 the following decomposition takes place:

Expressions using complex numbers

Consider the function of the complex variable z:
f (z) = z t.
Let us express the complex variable z in terms of the modulus r and the argument φ (r = |z|):
z = r e i φ .
We represent the complex number t in the form of real and imaginary parts:
t = p + i q .
We have:

Next, we take into account that the argument φ is not uniquely defined:
,

Let's consider the case when q = 0 , that is, the exponent - real number, t = p. Then
.

If p is an integer, then kp is an integer. Then, due to the periodicity of trigonometric functions:
.
That is, the exponential function with an integer exponent, for a given z, has only one value and is therefore unambiguous.

If p is irrational, then the products kp for any k do not produce an integer. Since k runs through an infinite series of values k = 0, 1, 2, 3, ..., then the function z p has infinitely many values. Whenever the argument z is incremented 2π(one turn), we move to a new branch of the function.

If p is rational, then it can be represented as:
, Where m, n- integers that do not contain common divisors. Then
.
First n values, with k = k 0 = 0, 1, 2, ... n-1, give n different meanings kp:
.
However, subsequent values ​​give values ​​that differ from the previous ones by an integer. For example, when k = k 0+n we have:
.
Trigonometric functions, whose arguments differ by values ​​that are multiples of 2π, have equal values. Therefore, with a further increase in k, we obtain the same values ​​of z p as for k = k 0 = 0, 1, 2, ... n-1.

Thus, an exponential function with a rational exponent is multivalued and has n values ​​(branches). Whenever the argument z is incremented 2π(one turn), we move to a new branch of the function. After n such revolutions we return to the first branch from which the countdown began.

In particular, a root of degree n has n values. As an example, consider the nth root of the real positive number z = x. In this case φ 0 = 0 , z = r = |z| = x, .
.
So, for a square root, n = 2 ,
.
For even k, (- 1 ) k = 1. For odd k, (- 1 ) k = - 1.
That is, the square root has two meanings: + and -.

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.

Lesson and presentation on the topic: "Power functions. Cubic root. Properties of the cubic root"

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Definition of a power function - cube root

Let's check the equality: $\sqrt((-x))$=-$\sqrt(x)$.
Let $\sqrt((-x))=a$ and $\sqrt(x)=b$. Let's raise both expressions to the third power. $–x=a^3$ and $x=b^3$. Then $a^3=-b^3$ or $a=-b$. Using the notation for roots we obtain the desired identity.

Properties of cube roots

Let's prove the second property. $(\sqrt(\frac(a)(b)))^3=\frac(\sqrt(a)^3)(\sqrt(b)^3)=\frac(a)(b)$.
We found that the number $\sqrt(\frac(a)(b))$ cubed is equal to $\frac(a)(b)$ and then equals $\sqrt(\frac(a)(b))$, which and needed to be proven.

Guys, let's build a graph of our function.
1) Domain of definition is the set of real numbers.
2) The function is odd, since $\sqrt((-x))$=-$\sqrt(x)$. Next, consider our function for $x≥0$, then display the graph relative to the origin.
3) The function increases when $x≥0$. For our function, a larger value of the argument corresponds to a larger value of the function, which means increase.
4) The function is not limited from above. In fact, from any large number we can calculate the third root, and we can go up to infinity, finding everything large values argument.
5) For $x≥0$ the smallest value is 0. This property is obvious.
Let's build a graph of the function by points at x≥0.

Let's construct our graph of the function over the entire domain of definition. Remember that our function is odd.

Function properties:
1) D(y)=(-∞;+∞).
2) Odd function.
3) Increases by (-∞;+∞).
4) Unlimited.
5) There is no minimum or maximum value.

7) E(y)= (-∞;+∞).
8) Convex downward by (-∞;0), convex upward by (0;+∞).

Examples of solving power functions

2. Construct a graph of the function. $y=\sqrt((x-2))-3$.
Solution. Our graph is obtained from the graph of the function $y=\sqrt(x)$, parallel transfer two units to the right and three units down.

3. Graph the function and read it. $\begin(cases)y=\sqrt(x), x≥-1\\y=-x-2, x≤-1 \end(cases)$.
Solution. Let's construct two graphs of functions on the same coordinate plane, taking into account our conditions. For $x≥-1$ we build a graph of the cubic root, for $x≤-1$ we build a graph of a linear function.
1) D(y)=(-∞;+∞).
2) The function is neither even nor odd.
3) Decreases by (-∞;-1), increases by (-1;+∞).
4) Unlimited from above, limited from below.
5) Greatest value No. Lowest value equals minus one.
6) The function is continuous on the entire number line.
7) E(y)= (-1;+∞).

Problems to solve independently